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Find the derivative of xaa + axa + aax

Find the derivative of the following function:

    \[ f(x) = x^{a^a} + a^{x^a} + a^{a^x}. \]


Let’s take the derivative of each term in the sum using the rule for the derivative of the exponential and the chain rule. For the first term, note that this is just x raised to some power so we can use the usual power rule,

    \[ \left( x^{a^a} \right)' = \left( a^a \right) \left( x^{a^a - 1} \right). \]

For the second term we need to use the formula for the derivative of a^x (which is (a^x)' = a^x \log a) and the chain rule,

    \[ \left( a^{x^a} \right)' = (x^a)' a^{x^a} \log a = ax^{a-1} a^{x^a} \log a. \]

For the final term we have

    \[ \left( a^{a^x} \right)' = (a^x)' a^{a^x} \log a = a^x a^{a^x} (\log a)^2. \]

Putting these all together we then have,

    \[ f'(x) = (a^a) (x^{a^a - 1} ) + ax^{a-1} a^{x^a} \log a + a^x a^{a^x} (\log a)^2. \]

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